When they needed to multiply large complicated numbers, they were smart and kept those numbers in the logarithm form. Then they could simply add those numbers. The same trick worked for multiplication.

Let's look at an example. I want to find volume of the earth.

volume = 4/3*pi * r^3 => 4.1888r^3
r = 6,378,100

I need to multiple a lot of numbers, 4.18888 and r three times to find the volume. Instead we'll use logarithms and addition:

This means that 68% (one standard deviation) of people in Seattle are 33 years old, plus or minus about 7 years.

So what's the probability on an 18 year old living in Seattle?

population of seattle = 620,778 #googled
population given age = f(age) * population of seattle
population given age(18) => 3,088.0792
population given age(30) => 33,278.3747

Hmm, there are only 3,000 18 year olds in Seattle - given my completely made up statistics above.

Why not modify those stats to match your home town?

Shannon's Entropy

Entropy of a variable is a measure of the uncertainty of its value.

H(X) = -sum(p(x)*ln(p(x)), x=X)

This function takes a set of values or messagesX and produces the entropy of that set. To do this, it needs to know the probability of a message being in X; this information is provided by the p function.

We can now calculate the entropy of the data given the various probabilities of messages:

H(data, p = pmax) => 1.0986
H(data, p = pmodel) => 0.9657

When the messages are most uncertain - they each have the same probability - the entropy is maxed out at about 1.1. If we make the number 11 most probable - 60% likelihood - then the entropy is only 0.95.

If we designate one of the messages as very likely to occur then the entropy should go down since there is very little uncertainty.

We need to make sure to specify the time with the units s, otherwise we'll get a bad result:

x(1) => -4.9m/s^2 + 100m/s + 490m

Units of Motion and Power

The velocity of an object is measured in meters m per second s.

velocity units = m / s

Its acceleration is the change in velocity over time:

accel units = velocity units / s

Isaac Newton tells us that all objects have a constant velocity (a = 0) unless acted upon by a force. This force is measured by multiplying the mass of the moving object (in kg) by the acceleration caused by the force. It's measured in, appropriately enough, Newtons:

N = kg * accel units => kg*m/s^2

If we apply a force over a distance (imagine pushing your car), then are said to have done work (so long as the object accelerates in the direction we're pushing!). Work is measured in Joules:

J = N*m => kg*m^2/s^2

There is another quantity called energy that is measured in Joules. Energy is the capacity for doing work. In every day speech, we use energy instead of work since work has a lot of real world connotations that are best left ignored.

The change in energy over time is called power and is measured in Watts.

W = J/s => kg*m^2/s^3

Voltage and Current:

V = J/C => kg*m^2/(s^2*C)
A = C/s => C/s
V*A => kg*m^2/s^3

Gravity

Let's start with Newton's Law that force is proportional to acceleration:

force = mass * accel

When I'm standing on earth, it is constantly accelerating me so that I stay glued to the ground. That acceleration is called standard gravity.